An adaptive digital equalizer is usually realized as a finite impulse response (FIR) filter, also known as a transversal filter. The filter output is obtained by linear combination of N signal samples stored in the equalizer delay line, i.e., EQU y.sub.n =C.sub.n X.sub.n.sup.T (1)
where C.sub.n ={c.sub.0,n, . . . , c.sub.N-1,n } is the vector of equalizer coefficients and X.sub.n ={x.sub.n, . . . , x.sub.n-N+1 } is the vector of signal samples stored in the equalizer delay line at lime n. The filter coefficients are traditionally adjusted by the least-mean square (LMS) algorithm with the objective of minimizing noise and residual signal distortion at the filter output. In the LMS algorithm, an error signal is first computed. This error signal is then correlated with each signal sample stored in the delay line to generate an estimate of the gradient vector of the mean-square error (MSE). The vector of filter coefficients is finally updated by subtracting a term that is proportional to that estimate. In reference-directed mode of operation, the error signal is obtained as the difference between the equalizer output signal and a known reference signal. In decision-directed mode, the reference signal is replaced by the locally determined, most likely received signal among the discrete signals of the employed signal constellation.
In reference-directed mode, convergence of the equalizer coefficients to correct settings can always be achieved, provided the receiver has proper knowledge of a training sequence sent prior to random data signals. If after this training, the receiver makes decisions with sufficiently low probability of error, equalizer adaptation will operate reliably also in decision-directed mode. However, if the equalizer is not well trained at the beginning of decision-directed equalization, convergence will usually not be achieved except in the cases of binary modulation or pure phase modulation.
The sending of a training sequence and the means required to determine in a receiver the exact time when reception of a training sequence begins introduce complexities in the modem design and lead to communication overhead, which may not be desirable. The use of a training sequence may also be inappropriate for various situations. For example, in multipoint networks of modems it is usually not practical to send training sequences from a master modem to the receivers of slave modems, which may independently be activated or deactivated by their users.
When initial reference-directed equalizer adjustment by a training sequence is either undesirable or not possible, self-training equalization methods must be employed to train an equalizer from random data signals. These methods generally require much longer training periods than needed for reference-directed training. However, for many applications the length of these training times is quite tolerable.
Self-training methods usually rely on the definition of a pseudo-error, which on average will lead to correct coefficient adjustment although initially no reliable individual decisions can be obtained from the equalizer output signals.
A partial-response class-IV (PRIV) system is defined by its discrete-time channel symbol response, which in D-transform notation is given by h.sub.PRIV (D)=1-D.sup.2, where D denotes the operator for delay by one modulation interval T. Note that in a PRIV system, intersymbol interference is introduced in a controlled fashion. This distinguishes PRIV systems from full-response systems, where no intersymbol interference is present. If a(D)= . . . a.sub.n D.sup.n +a.sub.n+1 D.sup.n+1 + . . . is the sequence of data symbols transmitted at the modulation rate 1/T, a sequence of correlated signal samples b(D)= . . . b.sub.n D.sup.n +b.sub.n+1 D.sup.n+1 = . . . (a.sub.n -a.sub.n-2)D.sup.n +(a.sub.n+1 -a.sub.n-1)D.sup.n+1 + . . . is obtained at the output of an ideal PRIV system. For example, for a quaternary PRIV system, the input symbols a.sub.n are taken from the set {-3, -1, +1, +3} and the ideal channel output signal samples can assume one of the seven levels {-6, -4, -2, 0, +2, +4, +6}. In general, for an M-ary system, the input symbols are taken from the set {-(M-1), . . . , -1, +1, . . . , +(M-1)} and the ideal channel output signal samples can assume one of the 2M-1 levels {-(2M-2), . . . , -2, 0, +2, . . . , +(2M-2)}. Note that max .vertline.a.sub.n .vertline.=M-1.
The objective of an adaptive digital equalizer for a PRIV system is to provide an equalized signal of the form EQU y.sub.n =(a.sub.n -a.sub.n-2)+e.sub.n =b.sub.n +e.sub.n (2)
where e.sub.n is an error signal due to noise and residual signal distortion. As mentioned above, in decision-directed mode, the equalizer coefficients are updated by the LMS algorithm EQU C.sub.n+1 =C.sub.n -.alpha..sub.dd e.sub.n X.sub.n (3)
where .alpha..sub.dd is the adaptation gain and e.sub.n is an estimate of the error e.sub.n. The estimate e.sub.n is given by EQU e.sub.n =y.sub.n -(a.sub.n -a.sub.n-2)=y.sub.n -b.sub.n (4)
where a.sub.n is a tentative decision on the transmitted symbol a.sub.n.
Self-training equalization is generally more difficult to achieve for partial-response systems than for full-reponse systems. This is especially the case for partial-response systems where the input signal alphabet comprises more than two signal levels. Since self-training is always a slow process, the degree to which the modulation rate and the phase of the received signal are recovered prior to equalization plays an important role. Self-training methods can easily fail when the phase of the received signal is unknown and can drift relative to the phase of the local receiver clock.
Self-training adaptive equalization has in the past mainly been applied for full-response systems, cf. the following publications: (a) D. N. Godard, "Self recovering equalization and carrier tracking in two-dimensional data communication systems", IEEE Trans. Commun., Vol. COM-28, pp. 1867-1875 (November 1980); (b) S. Bellini, "Bussgang techniques for blind equalization", Proc. of IEEE GLOBCOM 1986, pp. 46.1.1-46.1.7 (December 1986); and (c) G. Picchi et al., "Blind equalization and carrier recovery using a "Stop-and-Go" decision directed algorithm", IEEE Trans. Commun., Vol. COM-35, pp. 877-887 (September 1987).
Methods to achieve self-training equalization for partial-response systems have been proposed for linear and for distributed-arithmetic equalizers in following publications: (d) Y. Sato, "A Method of Self-Recovering Equalization for Multilevel Amplitude-Modulation Systems", IEEE Trans. Commun., Vol. COM-23, pp. 679-682 (June 1975); and (e) G. Cherubini, "Nonlinear Self-Training Adaptive Equalization for Partial-Response Systems", IEEE Trans. Commun., Vol. COM-42, pp. 367-376 (February 1994).
For example, the self-training algorithm proposed in prior art publication (d) mentioned above for computing the adjustments of the coefficients of a linear equalizer in a multilevel PRIV system can be expressed by EQU C.sub.n+1 =C.sub.n -.alpha.e.sub.S,n X.sub.n (5)
where .alpha.&gt;0 is the adaptation gain and e.sub.S,n is a pseudo-error. FIG. 1 shows an equivalent block diagram of this prior art self-training linear equalizer. The pseudo-error is generated by subtracting from the equalizer output y.sub.n a reconstructed PRIV signal b.sub.S,n, EQU e.sub.S,n =y.sub.n -b.sub.S,n. (6)
To obtain the signal b.sub.S,n, the equalizer output signal is first filtered by a filter with transfer characteristic 1/(1-.beta..sub.S D.sup.2), giving the signal EQU u.sub.S,n =y.sub.n +.beta..sub.S u.sub.S,n-2, (7)
where 0&lt;.beta..sub.s &lt;1. The transfer characteristic of this filter approximates the inverse of the transfer characteristic of an ideal PRIV channel, 1/(1-D.sup.2). Since the frequency response of a PRIV channel exhibits spectral nulls at 0 Hz and at the frequencies .+-.1/2T Hz, the parameter .beta..sub.s cannot be chosen equal to one. This choice would correspond to "inverting the channel" and would result in infinite noise enhancement.
The signal u.sub.S,n is input to a two-level decision element that generates the signal .gamma.sign(u.sub.S,n), where .gamma.=E{a.sub.n.sup.2 }/E{.vertline.a.sub.n .vertline.}. Finally, the reconstructed PRIV signal is obtained as EQU b.sub.S,n =.gamma..vertline.sign(u.sub.S,n)-sign(u.sub.S,n-2).vertline..(8)
Prior art solutions have the following disadvantages. The MSE achievable by a particular self-training algorithm depends on the method used to generate the pseudo-error. The methods for self-training adaptive equalization described in prior art publications (d) and (e) have the disadvantage that the variances of the employed pseudo-errors do not diminish below significant values even if the true MSE after equalization would become small. This means that the useful driving force provided by the pseudo-error relative to its random fluctuations is generally very small and that low MSE can only be achieved at the expense of very, very slow convergence. In partial-reponse systems, the problem of slow convergence is further aggravated by the fact that the received signals are highly correlated. Therefore the prior art methods can only be applied for systems where satisfactory timing recovery prior to equalization is achieved. In the case of partial response systems, however, no simple solution exists for timing recovery prior to equalizer convergence. In some communication systems, timing recovery in a conventional sense is not even performed. In these systems, an equivalent function is obtained by operating the receiver at fixed local timing and letting the equalizer coefficients adapt to changing timing phase. Thus, the equalizing methods proposed in prior art publications (d) and (e) mentioned above cannot be applied.